
In constructing a self-consistent kinetic equation one wants a system of field coordinates for which a simple linear substitution results in canonical coordinates. Otherwise one needs to express the wave basis in a Taylor series expansion in wave amplitude, and lack of clarity interpreting the broadened equations will ensue. Lagrangian field coordinates [1]}, [2]}, [3]} require a Taylor series expansion about the assumed smallness of the amplitude.

Attempts at deriving a self-consistent internal wave kinetic equation from the stand point of a stationary observer are doomed to failure for the same reason that afflicted [4]}: Doppler shifting of the small scales by the large. [5]} deals with this by effectively using only terms in the scattering cross-sections related to pressure and viscosity and thus, in some manner, references Langrangian estimates of correlation time scales.

One really wants to know the time scale \(\Gamma ^{-1}\)  (the inverse resonant bandwidth) because closure of the third order moment equation (REF ) ultimately rests upon an assumption that the time integration has converged. Substitution of possible relevant time scales suggests that convergence is problematic, but there are only guesses about the relevant time scales. See points #1 and #2 above.

